Varieties with definable factor congruences
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- by Pedro Sánchez Terraf and Diego J. Vaggione PDF
- Trans. Amer. Math. Soc. 361 (2009), 5061-5088 Request permission
Abstract:
We study direct product representations of algebras in varieties. We collect several conditions expressing that these representations are definable in a first-order-logic sense, among them the concept of Definable Factor Congruences (DFC). The main results are that DFC is a Mal’cev property and that it is equivalent to all other conditions formulated; in particular we prove that $\mathcal {V}$ has DFC if and only if $\mathcal {V}$ has $\vec {0}$ & $\vec {1}$ and Boolean Factor Congruences. We also obtain an explicit first-order definition $\Phi$ of the kernel of the canonical projections via the terms associated to the Mal’cev condition for DFC, in such a manner that it is preserved by taking direct products and direct factors. The main tool is the use of central elements, which are a generalization of both central idempotent elements in rings with identity and neutral complemented elements in a bounded lattice.References
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Additional Information
- Pedro Sánchez Terraf
- Affiliation: CIEM — Facultad de Matemática, Astronomía y Física (Fa.M.A.F.), Universidad Nacional de Córdoba, Ciudad Universitaria, Córdoba 5000, Argentina
- Email: sterraf@famaf.unc.edu.ar
- Diego J. Vaggione
- Affiliation: CIEM — Facultad de Matemática, Astronomía y Física (Fa.M.A.F.), Universidad Nacional de Córdoba, Ciudad Universitaria, Córdoba 5000, Argentina
- Email: vaggione@mate.uncor.edu
- Received by editor(s): December 15, 2006
- Published electronically: May 18, 2009
- Additional Notes: This work was supported by CONICET and SECYT-UNC
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 5061-5088
- MSC (2000): Primary 08B05; Secondary 03C40
- DOI: https://doi.org/10.1090/S0002-9947-09-04921-6
- MathSciNet review: 2515803